Local regression methods and systems for image processing systems

ABSTRACT

This disclosure provides methods, apparatus and systems for performing image processing regression for approximating multidimensional color transformation. According to an exemplary method, a shaping matrix is selected to minimize a cost function associated with a local linear regression representation of the color transformation.

BACKGROUND

Local linear regression is used in a variety of data fittingapplications. Particular applications within the realm of color imaginginclude printer and scanner characterization. A typical regressionproblem involves first gathering a training set of input data pointsfrom an input space and corresponding output data points from an outputspace. For the color characterization application, both input and outputspaces are multi-dimensional color spaces. The goal of the regressionalgorithm is then to derive mappings from every point in the input spaceto the output space while minimizing error over the training set. Anadditional consideration is to ensure that the regression does notoverfit the data in the sense that it is robust to noise in the trainingdata. Local regression algorithms are often used in situations where asingle global fit may be inadequate to approximate complex non-lineartransforms, as is typical in printer characterization. Instead, localtransforms are derived where the regression parameters vary as afunction of the input data point. Locality in regression is achieved byusing a weighting in the error minimization function which varies(typically decays) as a function of the distance from the regressiondata point. Choice of these weight functions is typically intuitivelyinspired, and not optimized for the training set. This sometimes resultsin large regression errors especially with sparse training data. A keyfundamental question hence remains on how to best use a certain localneighborhood of data points in regression problems.

INCORPORATION BY REFERENCE

R. Bala, “Device Characterization”, Digital Color Imaging Handbook,Chapter 5, CRC Press, 2003, is totally incorporated herein by referencein its entirety.

BRIEF DESCRIPTION

In one embodiment of this disclosure, a regression method forapproximating a multidimensional color transformation is disclosed whichcomprises (a) receiving a set Γ of training samples (x_(i), y_(i)),1≦i≦T, where x_(i) represents input color data to the multidimensionalcolor transformation, and y_(i) represents corresponding output colordata from the multidimensional color transformation; (b) selecting aparameterized form of a regression function f(x) that approximates themultidimensional color transformation; (c) receiving an input color x;(d) generating a cost function C representing a localized error producedby the regression function f(x) on the training set Γ, where thelocalized error is a function of both the parameters of f(x) and ashaping function that defines the shape and orientation of aneighborhood of training data localized around the input color x; (e)deriving the parameters of the regression function f(x) and shapingfunction to jointly minimize the cost function C; and (f) generating anoutput color y by calculating f(x) using the derived parameters of step(e).

In another aspect of this disclosure, a computer program product, thatwhen executed by a computer, causes the computer to execute a regressionmethod for approximating a multidimensional color transformation isdescribed. The method comprises (a) receiving a set Γ of trainingsamples (x_(i), y_(i)), 1≦i≦T, where x_(i) represents input color datato the multidimensional color transformation, and y_(i) representscorresponding output color data from the multidimensional colortransformation; (b) selecting a parameterized form of a regressionfunction f(x) that approximates the multidimensional colortransformation; (c) receiving an input color x; (d) generating a costfunction C representing a localized error produced by the regressionfunction f(x) on the training set Γ, where the localized error is afunction of both the parameters of f(x) and a shaping function thatdefines the shape and orientation of a neighborhood of training datalocalized around the input color x; (e) deriving the parameters of theregression function f(x) and shaping function to jointly minimize thecost function C; and (f) generating an output color y by calculatingf(x) using the derived parameters of step (e).

In still another aspect of this embodiment, an image processing methodis disclosed for rendering an image on an image output device. Themethod comprises receiving a device independent color spacerepresentation of the image; accessing an inverse characterizationtransform associated with the image output device to generate a devicedependent color space representation of the image, the inversecharacterization transform representing the inverse of amultidimensional color transformation associating a plurality of devicedependent color space values with a plurality of respective deviceindependent color space values, the multidimensional colortransformation generated by performing a method comprising: (a)receiving a set Γ of training samples (x_(i), y_(i)), 1≦i ≦T, wherex_(i) represents input color data to the multidimensional colortransformation, and y_(i) represents corresponding output color datafrom the multidimensional color transformation; (b) selecting aparameterized form of a regression function f(x) that approximates themultidimensional color transformation; (c) receiving an input color x;(d) generating a cost function C representing a localized error producedby the regression function f(x) on the training set Γ, where thelocalized error is a function of both the parameters of f(x) and ashaping function that defines the shape and orientation of aneighborhood of training data localized around the input color x; (e)deriving the parameters of the regression function f(x) and shapingfunction to jointly minimize the cost function C; and (f) generating anoutput color y by calculating f(x) using the derived parameters of step(e).

In yet another aspect of this embodiment, a computer program product isdisclosed that when executed by a computer, causes the computer toperform a color transformation for rendering an image on an image outputdevice. The method of deriving the color transformation comprises (a)receiving a set Γ of training samples (x_(i), y_(i)), 1≦i≦T, where x_(i)represents input color data to the multidimensional colortransformation, and y_(i) represents corresponding output color datafrom the multidimensional color transformation; (b) selecting aparameterized form of a regression function f(x) that approximates themultidimensional color transformation; (c) receiving an input color x;(d) generating a cost function C representing a localized error producedby the regression function f(x) on the training set Γ, where thelocalized error is a function of both the parameters of f(x) and ashaping function that defines the shape and orientation of aneighborhood of training data localized around the input color x; (e)deriving the parameters of the regression function f(x) and shapingfunction to jointly minimize the cost function C; and (f) generating anoutput color y by calculating f(x) using the derived parameters of step(e).

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow chart of an exemplary image processing regressionmethod according to this disclosure.

FIG. 2 is a plot of weight functions w(x, x₀) for x₀=(0,0), where theweight function decays as a function of distance from x₀ to generate aneighborhood around x₀.

FIG. 3 illustrates one exemplary function mapping from 2-D space (R²) to1-D space (R) to be approximated according to an exemplary regressionmethod according to this disclosure.

FIG. 4 illustrates the contours of the function illustrated in FIG. 3with exemplary training data overlaid.

FIG. 5 illustrates local regression using neighborhood shaping accordingto an exemplary embodiment of this disclosure.

FIG. 6 is a block diagram of an image processing system using anexemplary regression method according to this disclosure.

DETAILED DESCRIPTION

This disclosure provides methods and systems for local regression inderiving color transformations by introducing the notion of “shaping” inthe localizing weight function. The disclosed exemplary embodimentsinclude distinct features: 1.) a parameterization of the weight functiontypically used in local regression problems via a shaping matrix, and2.) a method to obtain the “optimal” shaping matrix by explicitlyintroducing the weight function parameters in the regression errormeasure. Demonstrated experimentally are that significant gains can bemade by optimizing “the shaping matrix” in local regression problems.Many color imaging applications including printer and scannercharacterization can benefit from the disclosed methods, apparatus andsystems. The disclosed exemplary embodiments are particularlyadvantageous for color devices that employ a large number of colorchannels, thus inducing a large dimensionality in the characterizationdata.

Regression is a common technique for estimating a functionalrelationship between input and output data, and is used frequently toderive color device characterization transformations. The lattertypically establish a functional relationship between a device dependentcolor space and a device independent color space. For printers, examplesof device dependent color spaces include CMY, CMYK or CMYKOV, where thesymbols stand for Cyan, Magenta, Yellow, Black, Orange, Violet,respectively. For display devices, the prevalent device dependent colorspace is RGB (or Red, Green, Blue). A common example of a deviceindependent color space is CIELAB. There are two types of colorcharacterization transforms—a forward and an inverse. For outputdevices, the forward transform maps a device dependent color space to adevice independent color space, and conversely, the inverse transformmaps a device independent color space to a device dependent color space.In certain applications, the forward transform for one device isconcatenated with the inverse transform for another device to produce a“device-to-device” color transformation. The regression techniques andexemplary embodiments described herein can be applied to forward,inverse, or device-to-device characterization transforms.

Linear regression is a specific case where the functional relationshipbetween the input and output spaces is approximated by a lineartransform. When the input and output data belong to multidimensionalvector spaces, the linear transform is a matrix. Specifically, considerthe problem where y in R^(m) is to be estimated as a function of aninput variable x in R^(n). Thus we have y≈f(x). Let Γ={(x_(i), y_(i)),i=1, 2, . . . , T} denote the set of training data over which thisresponse in known. The linear approximation is given by:y=f(x)=A.x, x ε R^(n) , y ε R ^(m) , A ε R ^(m×n)   (1)

The “best” regression parameter A is determined by minimizing theregression cost function that describes an aggregate error between y_(i)and Ax_(i) for the training set.

A variant of this is local linear regression as described in R. Bala,“Device Characterization,” Digital Color Imaging Handbook, Chapter 5.CRC Press, 2003, wherein the matrix A varies as a function of location xin input space. Thus we havey=f(x)=A_(x) .x, x ε R ^(n) , y ε R ^(m) , A ε R ^(m×n)   (2)

For each input data point x, the “best” regression parameter A_(x) isdetermined by minimizing the regression cost function:

$\begin{matrix}{{C\left( A_{x} \right)} = {\frac{1}{T}{\sum\limits_{i = 1}^{T}\;{{{y_{i} - {A_{x} \cdot x_{i}}}}^{2} \cdot {w\left( {x,x_{i}} \right)}}}}} & (3)\end{matrix}$

In the above cost function, note that it is the presence of weightfunction w(x, x_(i)) that introduces locality in regression.

The most general requirement is for w(x, x_(i)) to decay as a functionof the distance d(x, x_(i))=∥x−x_(i)∥. A popular instantiation is:w(x,x _(i))=e ^(−α)(∥x−x _(i)∥²)   (4)

The above weight function is plotted in FIG. 2 for a 2-D input variablex. Clearly, there is decay as a function of distance from input point xwhich means that in the cost function of Equation (3), more weight isattached to regression sets for which d(x, x_(i)) is small.Qualitatively, the subset of {x_(i)}'s for which w(x, x_(i)) is greaterthan a threshold (so as to have a significant impact on the costfunction) constitutes a local neighborhood of x.

Limitations of existing local linear regression is now discussed.

The use of such locality inducing weight functions is well-known tosignificantly help with regression accuracy over using a single “global”regression. The same has also been successfully applied to derive bothforward and inverse printer color transforms in R. Bala, “DeviceCharacterization”, Digital Color Imaging Handbook, Chapter 5, CRC Press,2003.

The notion of locality as in Equation (4) is meaningful from theviewpoint of neighborhood size, i.e. a certain a may be chosen tocontrol the spread of w(x, x_(i)) around x. That said, an importantconsideration that was previously ignored is shaping of w(x, x_(i)).

To appreciate this, consider, the 2-D function plotted in FIG. 3. FIG. 4shows contours of this function with training data overlaid. It isdesired to approximate this 2-D function with a locally linearregression function f( ), and to compute the regression output for theinput point labeled “x”.

It may be seen from FIG. 4 that many different regression neighborhoodsmay be defined in the vicinity of the input point. A few such circularneighborhoods are shown in FIG. 4. Note also that circles centered at apoint x₀ are contours of the distance function d(x, x₀)=∥x−x₀∥.

From a visual inspection of the plot in FIG. 3 it is clear that theoriginal 2-D function is simply linear in the vicinity of the inputpoint x, and can hence be perfectly approximated. But for that tohappen, the regression must pick or attach greater weight to trainingdata in the “linear region” vs. training data points in the non-linearregion. Note further that for the three different circular neighborhoodsshown in FIG. 4 the training data points identified as “x”s are closerto the input point, and will in fact get more weight, regardless of thesize (radius) of the circular neighborhoods.

Detailed next is how this problem may be averted by introducing thenotion of “neighborhood shaping.”

In this disclosure, utilized are the use of neighborhood shaping toaddress the problem discussed above. That is, the crucial observation ismade that for a fixed neighborhood size, shaping plays an important rolein regression accuracy.

Described now is how to achieve a desired neighborhood shaping. It isobserved that for finite-dimensional input spaces, the distance functiond(x, x_(i))=∥x−x_(i)∥ can alternatively be written as∥x−x _(i)∥²=(x−x _(i))^(T)(x−x _(i))

As shown in FIG. 4, contours of such a distance function result inhyper-spheres in R^(n) (special case circle in R²).

Proposed first is a generalization of this distance to:∥x−x _(i)∥_(Λ)=(x−x _(i))^(T)Λ(x−x _(i))where Λ is a positive definite matrix, which is a requirement to ensurenon-negativity of the distance metric for all x, x₀.

It is clear now that the contours of this new distance can begeneralized to be elliptical. A diagonal Λ with positive unequaldiagonal entries results in a hyper-ellipse with different ellipse radiiin different dimensions, while non-diagonal choices of Λ allow thecontrol of orientation.

Notably, the local linear transform A_(x) and the resulting outputestimates may vary considerably with different choices of Λ. Onepossible strategy to optimize Λ is to make it proportional to the samplecovariance matrix of the training data.

This disclosure and the exemplary embodiments described herein providemethods to find the best “shape” of the weight function or equivalentlyΛ for a fixed size/volume. To formally distinguish shape from size, were-write Λ as follows:Λ=λS ^(T) S; Λ, S ε R ^(n×n),where S denotes a “shape matrix” with determinant 1, and λ is anon-negative scalar relating to the size of the local neighborhood.

Given this separation of size and shape the shaping matrix may be solvedfor optimally by minimizing the regression error:

$\begin{matrix}{{C\left( {A_{x},\Lambda} \right)} = {\frac{1}{T}{\sum\limits_{i = 1}^{T}\;{{{y_{i} - {A_{x} \cdot x_{i}}}}^{2} \cdot {w\left( {x,x_{i}} \right)}}}}} & (5)\end{matrix}$where w(x, x_(i))=e^(−(x−x) ^(i) ⁾ ^(T) ^(Λ(x−x) ^(i) )subject to det(Λ)=cc=positive constant

Salient features of the optimization problem in Equation (5) are that(i) in this new setting, Λ or really the shape matrix S as well as theregression parameter matrix A_(x) are jointly optimized; and (ii) theconstraint c placed on the determinant of Λ fixes the size of theneighborhood.

Standard search-based constrained optimization techniques with asuitable choice of a starting point, can be used to determine theoptimum S and A_(x).

Note finally that although the embodiment and examples have beendescribed for linear regression, in principle the same technique readilyextends for nonlinear regression. The elements of the matrix A would besimply replaced by the parameters of the nonlinear approximationfunction.

Revisiting the regression problem in FIG. 3, the training data was fedto both classical local linear regression as well as to the proposedregression with neighborhood shaping. FIG. 5 visualizes the results. Inboth cases, for the same neighborhood size (appropriately parameterizedin either cost function) contours of the distance function are overlaidon the regression data. Clearly, in the proposed case, regression datapoints in the “linear region” afford more weight and hence theregression succeeds in perfectly approximating the function.

With reference to FIG. 1, illustrated is a flow chart of an exemplaryimage processing regression method incorporating neighborhood shaping asdiscussed hereto. By way of example, the method will be described withparticular reference to a printing system, however the image processingmethod is not limited to printing and can be applied to image processingin general.

In the context of a printing system, the image processing regressionmethod illustrated in FIG. 1 generates a multidimensional colortransformation associating a plurality of device dependent color spacevalues with a plurality of respective device independent color spacevalues.

To print a particular image, the inverse of the color transformationgenerated by the method of FIG. 1 is accessed to transform an imagerepresented in device independent color space to device, i.e. printer,dependent color space for rendering/printing on the printer. Aspreviously discussed, the device dependent color space representationsof the image , numerically indicate a relative amount of theircorresponding printer colors, e.g. CMYK, necessary to print the originalimage represented in device independent color space.

To generate the printer characterization transform, computer readableinstructions are executed in the following sequence:

Initially, the printer multidimensional color characterization transformgeneration algorithm starts 2.

Next, receiving a set Γ of training samples (x_(i), y_(i)), 1≦i≦T, isreceived where x_(i) represents input color data to the multidimensionalcolor transformation, and y_(i) represents corresponding output colordata from the multidimensional color transformation 4.

Next, a parameterized form of a regression function f(x) thatapproximates the multidimensional color transformation is selected 6.

Next, an input color x is received 8.

Next, a cost function C representing a localized error produced by theregression function f(x) on the training set Γ is generated 10, wherethe localized error is a function of both the parameters of f(x) and ashaping function that defines the shape and orientation of aneighborhood of training data localized around the input color x.

Next, the parameters of the regression function f(x) and shapingfunction to jointly minimize the cost function C are derived 12.

Next, an output color y is generated 14 by calculating f(x) using thederived parameters of the regression function f(x) and shaping functionto jointly minimize the cost function C.

Finally, the printer multidimensional color characterization transformgeneration algorithm ends 16.

With reference to FIG. 6, illustrated is a black diagram of an imageprocessing system using an exemplary regression method as discussedhereto and described with reference to FIG. 6.

In operation, the printing system receives a digital input 100,represented in device independent color space, and processes 102 thedevice independent color space representation of the digital input image100 to generate a pixel representation of the digital input imagesuitable for printing on printing device 106 to generate a hardcopyoutput 108 of the digital input image 100.

The image processing path 102 can reside and be executed on a DFE(Digital Front End), and/or the printing device 106. However, as will beunderstood by those of skill in the art, any computer related devicecapable of executing instructions can be used to process the image data.

As shown in FIG. 6, the image processing path includes amultidimensional color transformation, e.g. a look-up-table, whichincorporates data generated by the color transformation derivationmodule to produce device dependent color space representations of thedigital input image. Notably, the color transformation derivation moduleapproximates a multidimensional color transformation according to themethods described in this disclosure and specifically illustrated inFIG. 5.

After the digital input image is processed by the multidimensional colortransformation module to produce device dependent color spacerepresentations of the digital input image, the image data is processedaccording to specific tone reproduction curves 112 and halftoningalgorithms 114 to generate pixel data to be rendered on the printingdevice 106.

It will be appreciated that various of the above-disclosed and otherfeatures and functions, or alternatives thereof, may be desirablycombined into many other different systems or applications. Also thatvarious presently unforeseen or unanticipated alternatives,modifications, variations or improvements therein may be subsequentlymade by those skilled in the art which are also intended to beencompassed by the following claims.

What is claimed is:
 1. A regression method for approximating amultidimensional color transformation associated with an image outputdevice, the regression method executed by a controller operativelyconnected to the image output device, and the method comprising: (a)receiving a set Γ of training samples (x_(i), y_(i)), 1≦i≦T, where x_(i)represents input color data to the multidimensional colortransformation, and y_(i) represents corresponding output color datafrom the multidimensional color transformation; (b) selecting aparameterized form of a regression function f(x) that approximates themultidimensional color transformation; (c) receiving an input color x;(d) generating a cost function C representing a localized error producedby the regression function f(x) on the training set Γ, where thelocalized error is a function of both the parameters of f(x) and ashaping function that defines the shape and orientation of aneighborhood of training data localized around the input color x; (e)deriving the parameters of the regression function f(x) and shapingfunction to jointly minimize the cost function C; and (f) generating anoutput color y by calculating f(x) using the derived parameters of step(e).
 2. The method according to claim 1, wherein the shaping function instep (d) is a shaping matrix S.
 3. The method according to claim 2,wherein the regression function in step (b) is a local linear regressionfunction represented asy=f(x)=A _(x) .x, x ε R ^(n) , y ε R ^(m) , A εR ^(m×n).
 4. The methodaccording to claim 3, wherein the cost function C in step (d) is definedas${C = {{C\left( {A_{x},\Lambda} \right)} = {\frac{1}{T}{\sum\limits_{i = 1}^{T}\;{{{y_{i} - {A_{x} \cdot x_{i}}}}^{2} \cdot {w\left( {x,x_{i}} \right)}}}}}},$where w(x, x_(i))=e^(−(x−x) ^(i) ⁾ ^(T) ^(Λ(x−x) ^(i) ⁾, Λ=λS^(T)S,s isa shaping matrix, and the minimization of C is subject to the constraintthat the determinant of Λ=c, where c is a positive constant.
 5. Themethod of claim 1, wherein the multidimensional color transformation isa mapping from a device dependent color space to a device independentcolor space.
 6. A computer program product, that when executed by acomputer, causes the computer to execute a regression method forapproximating a multidimensional color transformation, the methodcomprising: (a) receiving a set Γ of training samples (x_(i), y_(i)),1≦i≦T, where x_(i) represents input color data to the multidimensionalcolor transformation, and y_(i) represents corresponding output colordata from the multidimensional color transformation; (b) selecting aparameterized form of a regression function f(x) that approximates themultidimensional color transformation; (c) receiving an input color x;(d) generating a cost function C representing a localized error producedby the regression function f(x) on the training set Γ, where thelocalized error is a function of both the parameters of f(x) and ashaping function that defines the shape and orientation of aneighborhood of training data localized around the input color x; (e)deriving the parameters of the regression function f(x) and shapingfunction to jointly minimize the cost function C; and (f) generating anoutput color y by calculating f(x) using the derived parameters of step(e).
 7. The computer program product according to claim 6, wherein theshaping function in step (d) is a shaping matrix S.
 8. The computerprogram product according to claim 7, the regression function in step(b) is a local linear regression function represented asy=f(x)=A _(x.) x, x ε R ^(n) , y ε R ^(m) , A ε R ^(m×n).
 9. Thecomputer program product according to claim 8, wherein the cost functionC in step (d) is defined as${C = {{C\left( {A_{x},\Lambda} \right)} = {\frac{1}{T}{\sum\limits_{i = 1}^{T}\;{{{y_{i} - {A_{x} \cdot x_{i}}}}^{2} \cdot {w\left( {x,x_{i}} \right)}}}}}},$where w(x, x_(i))=e^(−(x−x) ^(i) ⁾ ^(T) ^(Λ(x−x) ^(i)) Λ=λS^(T)S, S is ashaping matrix, and the minimization of C is subject to the constraintthat the determinant of Λ=c, where c is a positive constant.
 10. Thecomputer program product according to claim 6, wherein themultidimensional color transformation is a mapping from a devicedependent color space to a device independent color space.
 11. An imageprocessing method for rendering an image on an image output device, theimage processing method executed by a controller operatively connectedto the image output device, and the method comprising: receiving adevice independent color space representation of the image; accessing aninverse characterization transform associated with the image outputdevice to generate a device dependent color space representation of theimage, the inverse characterization transform representing the inverseof a multidimensional color transformation associating a plurality ofdevice dependent color space values with a plurality of respectivedevice independent color space values, the multidimensional colortransformation generated by performing a method comprising: (a)receiving a set Γ of training samples (x_(i), y_(i)), 1≦i≦T, where x_(i)represents input color data to the multidimensional colortransformation, and y_(i) represents corresponding output color datafrom the multidimensional color transformation; (b) selecting aparameterized form of a regression function f(x) that approximates themultidimensional color transformation; (c) receiving an input color x;(d) generating a cost function C representing a localized error producedby the regression function f(x) on the training set Γ, where thelocalized error is a function of both the parameters of f(x) and ashaping function that defines the shape and orientation of aneighborhood of training data localized around the input color x; (e)deriving the parameters of the regression function f(x) and shapingfunction to jointly minimize the cost function C; and (f) generating anoutput color y by calculating f(x) using the derived parameters of step(e).
 12. The image processing method according to claim 11, wherein theshaping function in step (d) is a shaping matrix S.
 13. The imageprocessing method according to claim 12, wherein the regression functionin step (b) is a local linear regression function represented asy=f(x)=A _(x) .x, x ε R ^(n) , y ε R ^(m) , A ε R ^(m×n).
 14. The imageprocessing method according to claim 13, wherein the cost function C instep (d) is defined as${C = {{C\left( {A_{x},\Lambda} \right)} = {\frac{1}{T}{\sum\limits_{i = 1}^{T}\;{{{y_{i} - {A_{x} \cdot x_{i}}}}^{2} \cdot {w\left( {x,x_{i}} \right)}}}}}},$where w(x, x_(i))=e^(−(x−x) ^(i) ⁾ ^(T) ^(Λ(x−x) _(is i)), Λ=λS^(T)S, Sis a shaping matrix, and the minimization of C is subject to theconstraint that the determinant of Λ=c, where c is a positive constant.15. The image processing method according to claim 11, wherein themultidimensional color transformation is a mapping from a devicedependent color space to a device independent color space.
 16. Acomputer program product that when executed by a computer, causes thecomputer to perform a color transformation for rendering an image on animage output device, the method of deriving the color transformationcomprising: (a) receiving a set Γ of training samples (x_(i), y_(i)),1≦i≦T, where x_(i) represents input color data to the multidimensionalcolor transformation, and y_(i) represents corresponding output colordata from the multidimensional color transformation; (b) selecting aparameterized form of a regression function f(x) that approximates themultidimensional color transformation; (c) receiving an input color x;(d) generating a cost function C representing a localized error producedby the regression function f(x) on the training set Γ, where thelocalized error is a function of both the parameters of f(x) and ashaping function that defines the shape and orientation of aneighborhood of training data localized around the input color x; (e)deriving the parameters of the regression function f(x) and shapingfunction to jointly minimize the cost function C; and (f) generating anoutput color y by calculating f(x) using the derived parameters of step(e).
 17. The computer program product according to claim 16, wherein theshaping function in step (d) is a shaping matrix S.
 18. The computerprogram product according to claim 17, wherein the regression functionin step (b) is a local linear regression function represented asy=f(x)=A _(x) .x, x ε R ^(n) , y ε R ^(m) , A ε R ^(m×n).
 19. Thecomputer program product according to claim 18, wherein the costfunction C in step (d) is defined as${C = {{C\left( {A_{x},\Lambda} \right)} = {\frac{1}{T}{\sum\limits_{i = 1}^{T}\;{{{y_{i} - {A_{x} \cdot x_{i}}}}^{2} \cdot {w\left( {x,x_{i}} \right)}}}}}},$where w(x, x_(i))=e^(−(x−x) ^(i) ⁾ ^(T) ^(A(x−x) ^(i)) , Λ=λS^(T)S, S isa shaping matrix, and the minimization of C is subject to the constraintthat the determinant of Λ=c, where c is a positive constant.
 20. Thecomputer program product according to claim 19, wherein themultidimensional color transformation is a mapping from a devicedependent color space to a device independent color space.